Okay, I now understand, however I have one question... Where the heck did the +8 come from?
You have
\[(9)3^{2k}-1\]
how did that become \[9(3^{2k}-1)+8\]
Actually, I understand @RaphaelFilgueiras
Is it much different? @satellite73 or am I just missing something in the working.

it is a trick of algebra, and the reason to use the trick is to make sure you can get \(3^{2k}-1\) out of \(3^{2k+2}-1\) so you can use the induction hypothesis
that is how i arrived at it to being with
i just wrote \(3^{2k}\times 9-1\) and forced \(3^{2k}-1\) out of it